Perfect fluid

In

mass density

\rho _{m}

and isotropic

In space-positive metric signature tensor notation, the stress–energy tensor of a perfect fluid can be written in the form

T^{\mu \nu }=\left(\rho _{m}+{\frac {p}{c^{2}}}\right)\,U^{\mu }U^{\nu }+p\,\eta ^{\mu \nu }\,

where U is the 4-velocity vector field of the fluid and where $\eta _{\mu \nu }=\operatorname {diag} (-1,1,1,1)$ is the metric tensor of

Minkowski spacetime

.

In time-positive metric signature tensor notation, the stress–energy tensor of a perfect fluid can be written in the form

T^{\mu \nu }=\left(\rho _{\text{m}}+{\frac {p}{c^{2}}}\right)\,U^{\mu }U^{\nu }-p\,\eta ^{\mu \nu }\,

where U is the 4-velocity of the fluid and where $\eta _{\mu \nu }=\operatorname {diag} (1,-1,-1,-1)$ is the metric tensor of

Minkowski spacetime

.

This takes on a particularly simple form in the rest frame

\left[{\begin{matrix}\rho _{e}&0&0&0\\0&p&0&0\\0&0&p&0\\0&0&0&p\end{matrix}}\right]

where $\rho _{\text{e}}=\rho _{\text{m}}c^{2}$ is the energy density and $p$ is the pressure of the fluid.

Perfect fluids admit a Lagrangian formulation, which allows the techniques used in field theory, in particular, quantization, to be applied to fluids.

Perfect fluids are used in

Friedmann–Lemaître–Robertson–Walker

equations to describe the evolution of the universe.

In general relativity, the expression for the stress–energy tensor of a perfect fluid is written as

T^{\mu \nu }=\left(\rho _{m}+{\frac {p}{c^{2}}}\right)\,U^{\mu }U^{\nu }+p\,g^{\mu \nu }\,

where U is the 4-velocity vector field of the fluid and where $g^{\mu \nu }$ is the inverse metric, written with a space-positive signature.

References

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